If $1,000 is deposited in a certain bank account and remains in the account along with any accumulated interest, the...

Understanding the Question

Let's clarify what we're asked to determine: Is the annual interest rate paid by the bank greater than 8 percent?

This is a yes/no question. We need to determine if \(\mathrm{r} > 8\), where r is the annual interest rate in percent.

Given Information

• Initial deposit: \(\$1,000\)
• Interest earned: \(\mathrm{I} = 1,000((1 + \mathrm{r}/100)^\mathrm{n} - 1)\)
• The deposit stays in the account with accumulated interest (compound interest)

What We Need to Determine

We need sufficient information to definitively answer whether \(\mathrm{r} > 8\) or \(\mathrm{r} \leq 8\). If we can determine this with certainty, the statement(s) will be sufficient.

Analyzing Statement 1

Statement 1 tells us: The deposit earns a total of $210 in interest in the first two years.

Testing the Threshold

Here's where strategic thinking saves time. Instead of solving for r exactly, let's test what would happen if \(\mathrm{r} = 8\%\):

If \(\mathrm{r} = 8\%\):

• After 2 years: \(\$1,000 \times (1.08)^2 = \$1,000 \times 1.1664 = \$1,166.40\)
• Interest earned: \(\$1,166.40 - \$1,000 = \$166.40\)

Key Comparison:

• Actual interest earned: \(\$210\)
• Interest at 8% rate: \(\$166.40\)

Since \(\$210 > \$166.40\), the actual interest rate must be greater than 8%.

Conclusion

We can definitively answer YES to the question: the annual interest rate IS greater than 8%.

[STOP - Sufficient!]

Statement 1 is SUFFICIENT.

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: \((1 + \mathrm{r}/100)^2 > 1.15\)

What This Means

This inequality tells us that the 2-year growth factor exceeds 1.15. In other words, the account grows by more than 15% over two years.

Testing the Boundary

Let's check if this constraint allows rates both above and below 8%:

Test 1: \(\mathrm{r} = 7.5\%\) (less than 8%)

• \((1 + 7.5/100)^2 = (1.075)^2 = 1.1556\)
• Since \(1.1556 > 1.15\) ✓, this rate satisfies the constraint
• But \(7.5\% < 8\%\), so the answer would be NO

Test 2: \(\mathrm{r} = 9\%\) (greater than 8%)

• \((1 + 9/100)^2 = (1.09)^2 = 1.1881\)
• Since \(1.1881 > 1.15\) ✓, this rate also satisfies the constraint
• And \(9\% > 8\%\), so the answer would be YES

Conclusion

Since the constraint allows both rates above 8% (giving YES) and rates below 8% (giving NO), we cannot definitively answer the question.

Statement 2 is NOT SUFFICIENT.

This eliminates choices B and D.

The Answer: A

Since Statement 1 alone is sufficient but Statement 2 alone is not sufficient, the answer is A.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."